Levels of inquiry: Hierarchies of pedagogical practices and inquiry processes

Provides pedagogical insight concerning the skill of inquiry The resource being annotated is: http://www.dlese.org/dds/catalog_COSEE-1808.htm

Superstrings on AdS3 at k=1

We study superstring theory in three dimensional Anti-de Sitter spacetime with NS-NS flux, focusing on the case where the radius of curvature is equal to the string length. This corresponds to the critical level k=1 in the Wess-Zumino-Witten description. Previously, it was argued that a transition takes place at this special radius, from a phase dominated by black holes at larger radius to one dominated by long strings at smaller radius. We argue that the infinite tower of modes that become massless at k=1 is a signal of this transition. We propose a simple two-dimensional conformal field theory as the holographic dual to superstring theory at k=1. As evidence for our conjecture, we demonstrate that at large N our putative dual exactly reproduces the full spectrum of the long strings of the weakly coupled string theory, including states unprotected by supersymmetry.Comment: 29 pages, one figure. An equivalent construction of the dual orbifold CFT has been added, together with a discussion of the short string spectrum and additional observations on interaction

On K(1)-local SU-bordism

This paper works towards a K(1)-local multiplicative splitting of SU-bordism

(k+1)-sums versus k-sums

A $k$-sum of a set $A\subseteq \mathbb{Z}$ is an integer that may be expressed as a sum of $k$ distinct elements of $A$. How large can the ratio of the number of $(k+1)$-sums to the number of $k$-sums be? Writing $k\wedge A$ for the set of $k$-sums of $A$ we prove that $$\frac{|(k+1)\wedge A|}{|k\wedge A|}\, \le \, \frac{|A|-k}{k+1}$$ whenever $|A|\ge (k^{2}+7k)/2$. The inequality is tight -- the above ratio being attained when $A$ is a geometric progression. This answers a question of Ruzsa.Comment: 5 page